Applications of dual MRM for determining the natural frequencies and natural modes of an Euler-Bernoulli beam using the singular value decomposition method

In this paper, a dual multiple reciprocity method (MRM) is employed to solv e the natural frequencies and natural modes for an Euler-Bernoulli beam. It is found that the conventional MRM using an essential integral equation re sults in spurious eigenvalues and modes. By using the natural integral equa tion of dual MRM, the spurious eigendata can be: filtered out. Four numeric al examples are given to verify the validity of the present formulation. In one of these four examples, fixed-fixed supported beam, it is found that t he boundary eigenvector cannot be determined by either the essential or nat ural integral equation alone since the rank of the corresponding leading co efficient matrix is insufficient. The singular value decomposition method i s then used to solve the eigenproblem after combining the essential and nat ural integral equations. This method can avoid the spurious eigenvalue prob lem and possible indeterminancy of boundary eigenvectors at the same time. (C) 1999 Elsevier Science Ltd. All rights reserved.